Arc measurement,[1] sometimes calleddegree measurement[2] (German:Gradmessung),[3] is theastrogeodetic technique of determining theradius of Earth and, byextension,its circumference. More specifically, it seeks to determine the localEarth radius of curvature of thefigure of the Earth, by relating thelatitude difference (sometimes also thelongitude difference) and thegeographic distance (arc length)surveyed between two locations on Earth's surface.[4] The most common variant involves onlyastronomical latitudes and themeridian arc length and is calledmeridian arc measurement; other variants may involve onlyastronomical longitude (parallel arc measurement) or bothgeographic coordinates (oblique arc measurement).[1]Arc measurement campaigns in Europe were the precursors to theInternational Association of Geodesy (IAG).[5]Nowadays, the method is replaced by worldwidegeodetic networks and bysatellite geodesy.
The first known arc measurement was performed byEratosthenes (240 BC) between Alexandria and Syene in what is now Egypt, determining the radius of the Earth with remarkable correctness. In the early 8th century,Yi Xing performed a similar survey.[6]
The French physicianJean Fernel measured the arc in 1528. The Dutch geodesistSnellius (~1620) repeated the experiment betweenAlkmaar andBergen op Zoom using more modern geodetic instrumentation (Snellius' triangulation).
Later arc measurements aimed at determining theflattening of the Earth ellipsoid by measuring at differentgeographic latitudes. The first of these was theFrench Geodesic Mission, commissioned by theFrench Academy of Sciences in 1735–1738, involving measurement expeditions to Lapland (Maupertuis et al.) and Peru (Pierre Bouguer et al.).
Friedrich Struve measured ageodetic control network viatriangulation between theArctic Sea and theBlack Sea, theStruve Geodetic Arc.
This is a partial chronological list of arc measurements:[7][8]
Assume theastronomic latitudes of two endpoints, (standpoint) and (forepoint) are known; these can bedetermined byastrogeodesy, observing thezenith distances of sufficient numbers ofstars (meridian altitude method).
Then, the empiricalEarth's meridional radius of curvature at the midpoint of the meridian arc can then be determined inverting thegreat-circle distance (orcircular arc length) formula:
where the latitudes are in radians and is thearc length onmean sea level (MSL).
Historically, the distance between two places has been determined at low precision bypacing orodometry.
High precision land surveys can be used to determine the distance between two places at nearly the same longitude by measuring abaseline and atriangulation network linkingfixed points. Themeridian distance from one end point to a fictitious point at the same latitude as the second end point is then calculated by trigonometry. The surface distance is reduced to the corresponding distance at MSL, (see:Geographical distance#Altitude correction).
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Additional arc measurements, at different latitudinal bands (each delimited by a new pair of standpoint and forepoint), serve todetermine Earth's flattening.Bessel compiled severalmeridian arcs, to compute the famousBessel ellipsoid (1841).Clarke (1858) combined most of the arc measurements then available to define a newreference ellipsoid.[9]
